Symmetry operators for 2nd quantized Hamiltonian

Symmetry operators in for second quantized operators only act on second quantized operators and do not act on numbers (i.e. the ‘first quantized Hamiltonian’, \mathcal{H}, or Hamiltonian density), UNLESS the symmetry operator is the time reversal operator, \mathcal{T}.

The symmetry operation, \mathcal{O}, is defined in terms of second quantized field operators: \mathcal{O}\psi\mathcal{O}^{-1} = U_{\mathcal{O}} \psi, where U_{\mathcal{O}} is a unitary matrix.

If a second quantized Hamiltonian, H=\Psi^\dag \mathcal{H}\Psi, is invariant under the symmetry operator, \mathcal{O}, then we have [H,\mathcal{O}]=0. By using the way the symmetry operation acts on the field operators, we can define an effective action of the symmetry operator on the Hamiltonian density, \mathcal{H}.

According to the above comment, it is trivial to write something like \mathcal{O}\mathcal{H}\mathcal{O}^{-1} since we always have \mathcal{O}\mathcal{H}\mathcal{O}^{-1}=\mathcal{H}, unless when \mathcal{O} is the time reversal operator \mathcal{T}, under which circumstance \mathcal{T}\mathcal{H}\mathcal{T}^{-1}=\mathcal{H}^*.


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